Journal of Nonlinear Mathematical Physics

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902 articles
Research Article

Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations

Rossen I Ivanov
Pages: 1 - 12
The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H 1 and H 1 right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a...
Research Article

Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves

David HENRY
Pages: 1 - 7
We show that within the framework of linear theory the particle paths in a periodic gravity-capillary or pure capillary deep-water wave are not closed.
Research Article

Complex Angle Variables for Constrained Integrable Hamiltonian Systems

S ABENDA, Yu FEDOROV
Pages: 1 - 4
We propose Dirac formalism for constraint Hamiltonian systems as an useful tool for the algebro-geometrical and dynamical characterizations of a class of integrable systems, the so called hyperelliptically separable systems. As a model example, we apply it to the classical geodesic flow on an ellipsoid.
Research Article

UV Manifold and Integrable Systems in Spaces of Arbitrary Dimension

A N LEZNOV
Pages: 1 - 7
The 2n dimensional manifold with two mutually commutative operators of differetiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.
Research Article

Lax Pairs, Painlevé Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2 + 1)-Dimensional Equation

Song-Ju YU, Kouichi TODA
Pages: 1 - 13
We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2 + 1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional...
Research Article

Kovalevski Exponents and Integrability Properties in Class A Homogeneous Cosmological Models

Marek SZYDLOWSKI, Marek BIESIADA
Pages: 1 - 10
Qualitative approach to homogeneous anisotropic Bianchi class A models in terms of dynamical systems reveals a hierarchy of invariant manifolds. By calculating the Kovalevski Exponents according to Adler - van Moerbecke method we discuss how algebraic integrability property is distributed in this class...
Research Article

Hidden Symmetries, First Integralsvand Reduction of Order of Nonlinear Ordinary Differential Equations

Barbara ABRAHAM-SHRAUNER
Pages: 1 - 9
The reduction of nonlinear ordinary differential equations by a combination of first integrals and Lie group symmetries is investigated. The retention, loss or even gain in symmetries in the integration of a nonlinear ordinary differential equation to a first integral are studied for several examples....
Research Article

A Chorin-Type Formula for Solutions to a Closure Model for the von K´arm´an­Howarth Equation 1

V N GREBENEV, M OBERLACK
Pages: 1 - 9
The article is devoted to studying the Millionshtchikov closure model (a particular case of a model introduced by Oberlack [14]) for isotropic turbulence dynamics which appears in the context of the theory of the von K´arm´an-Howarth equation. We write the model in an abstract form that enables us to...
Research Article

On a "Quasi" Integrable Discrete Eckhaus Equation

M J ABLOWITZ, C D AHRENS
Pages: 1 - 12
In this paper, a discrete version of the Eckhaus equation is introduced. The discretiztion is obtained by considering a discrete analog of the transformation taking the cotinuous Eckhaus equation to the continuous linear, free Schrödinger equation. The resulting discrete Eckhaus equation is a nonlinear...
Research Article

From Bi-Hamiltonian Geometry to Separation of Variables: Stationary Harry-Dym and the KdV Dressing Chain

Maciej BLASZAK
Pages: 1 - 13
Separability theory of one-Casimir Poisson pencils, written down in arbitrary coordnates, is presented. Separation of variables for stationary Harry-Dym and the KdV dressing chain illustrates the theory.
Research Article

On a graded q-differential algebra

Viktor ABRAMOV
Pages: 1 - 8
Given an associative unital ZN -graded algebra over the complex numbers we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-differential d of the graded q-differential algebra is a homogeneous endomorphism of degree 1 satisfying...
Research Article

Noncentral Extensions as Anomalies in Classical Dynamical Systems

Jorge E SOLOMIN, Marcela ZUCCALLI
Pages: 1 - 9
A two cocycle is associated to any action of a Lie group on a symplectic manifold. This allows to enlarge the concept of anomaly in classical dynamical systems considered by F Toppan in [J. Nonlinear Math. Phys. 8, Nr. 3 (2001), 518­533] so as to encompass some extensions of Lie algebras related to noncanonical...
Research Article

Harmonic maps between quaternionic Kahler manifolds

S. Ianus, R. Mazzocco, G.E. Vilcu
Pages: 1 - 8
Research Article

On the Quantization of Yet Another Two Nonlinear Harmonic Oscillators

Francesco CALOGERO
Pages: 1 - 6
In two previous papers the quantization was discussed of three one-degree-of-freedom Hamiltonians featuring a constant c, the value of which does not influence at all the corresponding classical dynamics (which is characterized by isochronous solutions, all of them periodic with period 2: "nonlinear...
Research Article

Solutions of WDVV Equations in Seiberg-Witten Theory from Root Systems

R. MARTINI, P.K.H. GRAGERT
Pages: 1 - 4
We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed.
Research Article

A Note on Surface Profiles for Symmetric Gravity Waves with Vorticity

Mats EHRNSTROM
Pages: 1 - 8
We consider a nontrivial symmetric periodic gravity wave on a current with nondcreasing vorticity. It is shown that if the surface profile is monotone between trough and crest, it is in fact strictly monotone. The result is valid for both finite and infinite depth.
Research Article

Asymptotic behavior of discrete holomorphic maps zc and log(z)

Sergey I AGAFONOV
Pages: 1 - 14
It is shown that discrete analogs of zc and log(z), defined via particular "integrable" circle patterns, have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painlevé-II equations, asymptotics of these solutions providing...
Research Article

Nonlocal Extensions of Similarity Methods

George Bluman
Pages: 1 - 24
Similarity methods include the calculation and use of symmetries and conservation laws for a given partial differential equation (PDE). There exists a variety of software to calculate and use local symmetries and local conservation laws. However, it is often the case that a given PDE admits no useful...
Research Article

Quantum Integrability of the Dynamics on a Group Manifold

V Aldaya, M Calixto, J Guerrero, F F Lopez-Ruiz
Pages: 1 - 12
We study the dynamics of a particle moving on the SU(2) group manifold. An exact quantization of this system is accomplished by finding the unitary and irreducible representations of a finite-dimensional Lie subalgebra of the whole Poisson algebra in phase space. In fact, the basic position and momentum...
Research Article

Navier­Stokes Equations with Nonhomogeneous Dirichlet Data

Herbert AMANN
Pages: 1 - 11
We discuss the solvability of the time-dependent incompressible Navier­Stokes equtions with nonhomogeneous Dirichlet data in spaces of low regularity.
Research Article

Integrability of the Perturbed KdV Equation for Convecting Fluids: Symmetry Analysis and Solutions

J.M. CERVER´O, O. Zurr'on
Pages: 1 - 23
As an example of how to deal with nonintegrable systems, the nonlinear partial differential equation which describes the evolution of long surface waves in a convecting fluid ut + (uxxx + 6uux) + 5uux + (uxxx + 6uux)x = 0, is fully analyzed, including symmetries (nonclassical and contact transformatons),...
Review Article

So. . . what was the question?

Gérard G EMCH
Pages: 1 - 8
An overview of the lectures at the 2002 Bialowiea Workshop is presented. The symbol* after a proper name indicates that a copy of the corresponding contribution to the proceedings was communicated to the author of this summary.
Research Article

Comparisons Between Vector and Matrix Padé Approximants

Claude BREZINSKI
Pages: 1 - 12
In this paper, we compare the degrees and the orders of approximation of vector and matrix Padé approximants for series with matrix coefficients. It is shown that, in this respect, vector Padé approximants have better properties. Then, matrix­vector Padé approximants are defined and constructed. Finally,...
Research Article

On the Moyal Quantized BKP Type Hierarchies

Dolan Chapa SEN, A. Roy CHOWDHURY
Pages: 1 - 7
Quantization of BKP type equations are done through the Moyal bracket and the formalism of pseudo-differential operators. It is shown that a variant of the dressing operator can also be constructed for such quantized systems.
Research Article

Non-linear Schrödinger Equations, Separation and Symmetry

George SVETLICHNY
Pages: 2 - 26
We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to...
Research Article

Neumann and Bargmann Systems Associated with an Extension of the Coupled KdV Hierarchy

Zhimin JIANG
Pages: 5 - 12
An eigenvalue problem with a reference function and the corresponding hierarchy of nonlinear evolution equations are proposed. The bi-Hamiltonian structure of the hierarchy is established by using the trace identity. The isospectral problem is nonlinearized as to be finite-dimensional completely integrable...
Research Article

Taming Spatiotemporal Chaos by Impurities in the Parametrically Driven Damped Nonlinear Schrödinger Equation

N V ALEXEEVA, I V BARASHENKOV, G P TSIRONIS
Pages: 5 - 12
Solitons of the parametrically driven, damped nonlinear Schrödinger equation become unstable and seed spatiotemporal chaos for sufficiently large driving amplitudes. We show that the chaos can be suppressed by introducing localized inhomogeneities in the parameters of the equation. The pinning of the...
Research Article

Gauge Transformations for a Family of Nonlinear Schrödinger Equations

Gerald A. GOLDIN
Pages: 6 - 11
An enlarged gauge group acts nonlinearly on the class of nonlinear Schrödinger equations introduced by the author in joint work with Doebner. Here the equations and the group action are displayed in the presence of an external electromagnetic field. All the gauge-invariants are listed for the coupled...
Research Article

µ-Holomorphic Projective Connections and Conformal Covariance

Mohamed KACHKACHI
Pages: 7 - 12
At the quantum level of a bidimensional conformal model, the conformal symmtry is broken by the diffeomorphism anomaly and the conformal covariance is not maintained. Here we interpret geometrically this conformal covariance as an exact holomorphy condition on a two-dimensional Riemann surface on which...
Research Article

Correctors for Some Nonlinear Monotone Operators

Johan BYSTRÖM
Pages: 8 - 30
In this paper we study homogenization of quasi-linear partial differential equations of the form -div (a (x, x/h, Duh)) = fh on with Dirichlet boundary conditions. Here the sequence (h) tends to 0 as h and the map a (x, y, ) is periodic in y, monotone in and satisfies suitable continuity conditions....
Research Article

An invariant p-adic q-integral associated with q-Euler numbers and polynomials

Ismail Naci CANGÜL, Veli KURT, Yilmaz SIMSEK, Hong Kyung PAK, Seog-Hoon RIM
Pages: 8 - 14
The purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1, 3, 6, 12], and to derive the "complete sums for q-Euler polynomials" which are evaluated by using...
Research Article

Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

M.L. GANDARIAS, M.S. BRUZON
Pages: 8 - 12
We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the...
Research Article

A Note on q-Bernoulli Numbers and Polynomials

A S HEGAZI, M MANSOUR
Pages: 9 - 18
In this paper, we define a new q-analogy of the Bernoulli polynomials and the Bernoulli numbers and we deduced some important relations of them. Also, we dduced a q-analogy of the Euler-Maclaurin formulas. Finally, we present a relation between the q-gamma function and the q-Bernoulli polynomials.
Research Article

Noetherian first integrals

PGL LEACH, GP FLESSAS
Pages: 9 - 21
From time to time one finds claims in the literature that first integrals/invariants of Lagrangian systems are nonnoetherian. Such claims diminish the contribution of Noether in the topic of integrability. We provide an explicit demonstration of noethe- rian symmetries associated with the integrals which...
Research Article

Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

Harald GROSSE, Raimar WULKENHAAR
Pages: 9 - 20
We first review regularization methods based on matrix geometry which provide an ultraviolet cut-off for scalar fields respecting the symmetries. Sections of bundles over the sphere can be quantized, too. This procedure even allows to regularize supesymmetry without violating it. Recently, this work...
Research Article

Geometric approach to BRST-symmetry and ZN-generalization of superconnection

V ABRAMOV, O LIIVAPUU
Pages: 9 - 20
We propose a geometric approach to the BRST-symmetries of the Lagrangian of a topological quantum field theory on a four dimensional manifold based on the formalism of superconnections. Making use of a graded q-differential algebra, where q is a primitive N-th root of unity, we also propose a notion...